MAT 150A Final: MAT150A_DAY19_Nov28_Lecture

• School: UC-Davis
• Department: Mathematics
• Course: MAT 150A
• Professor: José Simental Rodríguez

Document Summary

Example nonfaithful group action g = sn, x = r g x = (g)x (12)x = x, (123) x = x multiply by sign of the permutation. If g x = x x r, we can"t conclude that g = e. : d6 s3 | ker( )| = 2. 6(cid:17)) = (12) conjugation action g c x = gxg 1. Let g g be st g c x = x x g. Gx = xg x g g z(g). The conjugation action is faithful i z(g) = {e} e. g. the conjugation action of dn is faithful i n is odd. The conjugation action is trivial (f c x = x g x) i g is abelian. Let g be a nite group, and consider the conjugation action. By the counting formula, for each x g, |ox| divides |g| x g. The class eqn : |g| = 1 + xo orbits. O 6= {e} each one of these divides |g|.